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<div><i><font color="Green">E34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="qc_lang1.html#NM6">QC-pred_symbol</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="qc_lang1.html#M1" target="_self">QC-variable_list</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:41_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="qc_lang1.html#K9" target="_self">!</a> <font color="Maroon">c<sub>3</sub></font> is <a href="qc_lang1.html#V2" target="_self">atomic</a>
 <b>by </b><i><a class="ref" href="qc_lang1.html#D17" target="_self">Def17</a></i>;<br/><b>hence </b><a NAME="E2:41_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font> <a href="qc_lang1.html#K9" target="_self">!</a> <font color="Maroon">c<sub>3</sub></font>]
 <b>by </b><i><a class="txt" href="qc_lang1.html#E1:41"><i><font color="Green">E33</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:41_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[ <a href="qc_lang1.html#K11" target="_self">VERUM</a> ]
 
<b>by </b><i><a class="txt" href="qc_lang1.html#E1:41"><i><font color="Green">E33</font></i></a></i>;<br/>
<div><i><font color="Green">E36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> ;<br/><b>assume </b><a NAME="E1:41_1_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font>]
 ;<br/><a NAME="E2:41_1_2"/><i><font color="Green">E38</font></i>: 
 <a href="qc_lang1.html#K12" target="_self">'not'</a> <font color="Maroon">c<sub>1</sub></font> is <a href="qc_lang1.html#V3" target="_self">negative</a>
 <b>by </b><i><a class="ref" href="qc_lang1.html#D18" target="_self">Def18</a></i>;<br/><a NAME="E3:41_1_2"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K17" target="_self">the_argument_of</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K12" target="_self">'not'</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="qc_lang1.html#D23" target="_self">Def23</a></i>;<br/><b>hence </b><a NAME="E4:41_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[ <a href="qc_lang1.html#K12" target="_self">'not'</a> <font color="Maroon">c<sub>1</sub></font>]
 <b>by </b><i><a class="txt" href="qc_lang1.html#E1:41"><i><font color="Green">E33</font></i></a>, <a class="txt" href="qc_lang1.html#E1:41_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="qc_lang1.html#E2:41_1_2"><i><font color="Green">E38</font></i></a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E37</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> ;<br/><b>assume </b><a NAME="E1:41_1_3"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>] )
 ;<br/><a NAME="E2:41_1_3"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="qc_lang1.html#K13" target="_self">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font> is <a href="qc_lang1.html#V4" target="_self">conjunctive</a>
 <b>by </b><i><a class="ref" href="qc_lang1.html#D19" target="_self">Def19</a></i>;<br/><a NAME="E3:41_1_3"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K18" target="_self">the_left_argument_of</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="qc_lang1.html#K13" target="_self">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K19" target="_self">the_right_argument_of</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="qc_lang1.html#K13" target="_self">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="qc_lang1.html#D24" target="_self">Def24</a>, <a class="ref" href="qc_lang1.html#D25" target="_self">Def25</a></i>;<br/><b>hence </b><a NAME="E4:41_1_3"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font> <a href="qc_lang1.html#K13" target="_self">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font>]
 <b>by </b><i><a class="txt" href="qc_lang1.html#E1:41"><i><font color="Green">E33</font></i></a>, <a class="txt" href="qc_lang1.html#E1:41_1_3"><i><font color="Green">E38</font></i></a>, <a class="txt" href="qc_lang1.html#E2:41_1_3"><i><font color="Green">E39</font></i></a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> ;<br/><b>assume </b><a NAME="E1:41_1_4"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>]
 ;<br/><a NAME="E2:41_1_4"/><i><font color="Green">E40</font></i>: 
 <a href="qc_lang1.html#K14" target="_self">All</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> is <a href="qc_lang1.html#V5" target="_self">universal</a>
 <b>by </b><i><a class="ref" href="qc_lang1.html#D20" target="_self">Def20</a></i>;<br/><a NAME="E3:41_1_4"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K21" target="_self">the_scope_of</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K14" target="_self">All</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="qc_lang1.html#D27" target="_self">Def27</a></i>;<br/><b>hence </b><a NAME="E4:41_1_4"/>
<font color="Maroon">P<sub>1</sub></font>[ <a href="qc_lang1.html#K14" target="_self">All</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>]
 <b>by </b><i><a class="txt" href="qc_lang1.html#E1:41"><i><font color="Green">E33</font></i></a>, <a class="txt" href="qc_lang1.html#E1:41_1_4"><i><font color="Green">E39</font></i></a>, <a class="txt" href="qc_lang1.html#E2:41_1_4"><i><font color="Green">E40</font></i></a></i>;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E6:41_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a>  holds <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="qc_lang1.html#S1" target="_self">QC_LANG1:sch 1</a>(<a class="txt" href="qc_lang1.html#E1:41_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="qc_lang1.html#E2:41_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="qc_lang1.html#E3:41_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="qc_lang1.html#E4:41_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="qc_lang1.html#E5:41_1"><i><font color="Green">E38</font></i></a>);</i><br/>


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